Also a somewhat biased opinion, but if you're going to learn to play a League of Legends / Dota style game, pick Dota2.
One thing you must be vigilant with on PC gaming is the "free to play" games that can wind up costing you a fortune.
For my own kids when they first started out, I gave them a lot of leeway, and they wound up spending large sums of money on Team Fortress 2 cosmetics/guns.
League of Legends, while technically free to play, requires that you pay cash to unlock gameplay options, or sink hundreds and hundreds of hours into unlocking them by grinding in-game points. There is a pool of characters every week that is free, but you pay money to gain access to more. Dota2 is truly free to play in the sense that you get ALL of the gameplay content for free, but you can pay money for costumes and compendiums, neither of which have any impact on core gameplay. Different pricing models affect the gaming experience as well. Dota heroes are all unique. On paper, ALL of them are "overpowered" in some way, and this works because there are 110+ options to choose from, so counterplay is possible. In League of Legends, because many people might not have access to the entire hero pool, there is a lot of homogenization. If the free hero pool of the week contained a lot of "paper" heroes, it wouldn't be fair if one of the pay-to-play guys had access to scissors and just tore everyone up. It's been years since I've touched League, so maybe improvements have been made on that front. A quick Google search says that if you bought all the available Heroes, it would run you more than $700. That's a lot of scratch to gain access to all of the in-game content in a "free to play" game.
There are a lot of other PC games that follow the League of Legends model. It's about being an educated consumer.
So while it's fine that my kids wound up spending a bunch of money on TF2 hats, I've tried to get them into the mindset of instead using their funds to expand their gaming horizons. My 14 year old has an extensive Steam wishlist, and gets e-mail notifications when things go on sale. He's got dozens and dozens of games now, and has picked many of them up for deep discounts. My daughter is perfectly content playing a ton of Minecraft and piggybacking on my library for the most part. The eldest is 18 now and still mostly plays TF2. To each his own.
If you are interested in playing old arcade and console titles, my personal suggestion is to use RetroArch. It's extremely powerful, albeit moderately difficult to set up.
There are online tutorials available.
PCs do more than gaming, though!
Almost a decade ago I dumped TV and went to a full streaming household. I have Home Theater PCs attached to all the various TVs in the house, each with access to a central media server. Instead of a $175/mo cable bill, I pay $60 for the best internet package, and spent $250-400 on PCs for each TV. I overpaid for the boxes. You can do a lot with $35 Raspberry Pis these days. All of those boxes are capable of streaming 1080 video content from the web, but they also have access to thousands of movies and tens of thousands of hours of TV shows on my Media server. No commercials, super easy-to-use interface. Kids watch Netflix in the living room, or just use a wireless mouse/KB to navigate YouTube. Each TV has access to the ROM library as well. I use Wireless Xbox360 controllers in most of the house, but a Steam Controller in my bedroom. I can load up Super Mario Bros 3 in the living room, save it, then go upstairs to the bedroom and start right where I left off. I spent a little extra on the box I have in my bedroom to be able to play many of my Steam titles from bed at decent settings, but with Steam in-home streaming, I think that can also be done with a much less expensive machine. The
In-home streaming lets me use the horsepower of my primary gaming PC in the office and stream the video/controls right to the TV in my bedroom at quality levels higher than what the actual box in the bedroom is capable of. I played a lot of Xcom this way, but I also play platformers/twin stick games from bed. The TV is too far away to read small text, and bed isn't really the place for a mouse/keyboard. There's no noticeable input lag, but I'm sure the graphics take a small hit.
In short.... **** consoles. Honestly. Also, there simply aren't enough hours in the day.